Let $n$ be a non-zero natural number. Let $f$ be in $\mathcal{S}_n$. For $k$ in $\llbracket 1, 2n \rrbracket$, we denote $\varphi_k = \frac{\pi}{2n} + \frac{k\pi}{n}$ and $\omega_k = \mathrm{e}^{\mathrm{i}\varphi_k}$. Verify that, for all $k \in \llbracket 1, 2n \rrbracket$, $\frac{2\omega_k}{(1 - \omega_k)^2} = \frac{-1}{2\sin(\varphi_k/2)^2}$ and deduce from questions 11 and 12 that $$\forall \theta \in \mathbb{R}, \quad f'(\theta) = \frac{1}{2n} \sum_{k=1}^{2n} f(\theta + \varphi_k) \frac{(-1)^k}{2\sin(\varphi_k/2)^2}.$$
Let $n$ be a non-zero natural number. Let $f$ be in $\mathcal{S}_n$. For $k$ in $\llbracket 1, 2n \rrbracket$, we denote $\varphi_k = \frac{\pi}{2n} + \frac{k\pi}{n}$ and $\omega_k = \mathrm{e}^{\mathrm{i}\varphi_k}$.
Verify that, for all $k \in \llbracket 1, 2n \rrbracket$, $\frac{2\omega_k}{(1 - \omega_k)^2} = \frac{-1}{2\sin(\varphi_k/2)^2}$ and deduce from questions 11 and 12 that
$$\forall \theta \in \mathbb{R}, \quad f'(\theta) = \frac{1}{2n} \sum_{k=1}^{2n} f(\theta + \varphi_k) \frac{(-1)^k}{2\sin(\varphi_k/2)^2}.$$